
Assume \(T_0\)  extends \(\mathsf {ZFC}+\)there are class many Woodin cardinals which are a limit of Woodin cardinals.
Then the \(\in \) -formula \(\phi _0(x)\)  definingThe Chang model \(L(\mathrm {Ord}^\omega )\)  is the smallest transitive model of \(\mathsf {ZF}\)  containing all the countable sequences of ordinals. \(\mathcal {P}\left(2^\omega \right)^{L(\mathrm {Ord}^\omega )}\)  defines a generically tame family for \(T_0\) ; e.g. \(T_0\models {\mathbf {MAX}(\mathsf {\mathcal {A}})}\)  for \(\mathcal {A}=\mathcal {P}\left(2^\omega \right)^{L(\mathrm {Ord}^\omega )}\) .

Assume \(T_1\)  extends \(T_0\)  with the axiomThis is first order expressible by [1]}.
Any model \((V,\in )\)  of \(T_1\)  is a generic extension of some inner model \((W,\in )\)  where some countable ordinal \(\delta \)  of \(V\)  is a supercompact cardinal in
\(W\) .
Then the universally Baire sets
are a generically tame family for \(T_1\) ; e.g. \(T_1\models {\mathbf {MAX}(\mathsf {UB})}\)  for \(\phi _\text{{\sf UB}}(x)\)  an \(\in \) -formula defining the universally Baire sets.

